Table 2.

A summary of key mathematical modeling approaches in cancer nanomedicine

Biological problem	Modeling-type	References	Major findings	Clinical relevance
Biomolecular corona formation	Kinetic modeling	(Dell'Orco et al. 2010; Dell'Orco et al. 2012; Sahneh et al. 2013)	During corona formation, high affinity proteins displace low affinity proteins (Vroman effect), and the corona evolves from a metastable to a stable state. NP size is more important than number and size of peptides bound to NP surface in governing successful NP-cell surface receptor binding.	Provide a framework to study microscopic nano-bio interactions in various physiological conditions.
	Coarse-grained molecular dynamics simulations	(Lopez and Lobaskin 2015; Tavanti et al. 2015a)	Protein adsorption energies for NP-protein interaction are primarily affected by NP size, while surface charge only has a small effect.
Microvascular transport, margination, and binding	Continuum modeling	(Gentile et al. 2008)	An increase in hematocrit or vessel permeability reduces the effective diffusion coefficient of NPs, highlighting implications to intravascular transport of NPs.	Provide insights into capillary-scale biophysical interactions of NPs that can impact their macroscopic behavior, thereby providing design guidelines to optimize systemic circulation kinetics.
		(Tsoi et al. 2016)	NP sequestration in liver sinusoid is jointly affected by hemodynamic conditions and NP characteristics.
	Hybrid modeling	(Lee et al. 2013; Müller et al. 2014; Fullstone et al. 2015)	Larger NP size correlates with greater margination, which is further promoted by discoidal NP shape and higher hematocrit.
Cellular internalization	Discrete modeling	(Gao et al. 2005; Decuzzi and Ferrari 2008; Yuan and Zhang 2010)	A minimal particle size and ligand density are necessary for effective endocytosis.	Provide mechanistic understanding of the cellular uptake of NPs, which has implications in drug delivery or NP clearance by immune cells.
Whole-body biodistribution and clearance	PK modeling	(Dogra et al. 2018)	Small NP size correlates with longer systemic circulation and lower accumulation in mononuclear phagocytic system (MPS) organs, irrespective of route of injection. Positive charge supports excretion, and surface exposure of charged molecules increases the vulnerability to sequestration in MPS organs.	Provide a mechanistic description of whole-body phenomenological observations important for quantifying structure-activity relationships of NPs.
Tumor deliverability	Hybrid modeling	(Chauhan et al. 2012; Hendriks et al. 2012; Frieboes et al. 2013; Stapleton et al. 2013; Sykes et al. 2016)	Interplay between NP physicochemical properties (especially, size, surface charge) and vascular characteristics affects EPR-based accumulation and delivery of NPs to cancerous cells in the tumor interstitium.	These models provide insight on the intra-tumoral transport of NPs and provide critical design guidelines for improved tumor deliverability.
Nanotherapy efficacy and toxicity	PD modeling	(Pascal et al. 2013a; Leonard et al. 2016; Wang et al. 2016; Miller and Frieboes 2018)	Time integrated NP uptake by cancerous cells governs therapy efficacy. The outcome however is non-trivially affected by patient-specific tumor-perfusion heterogeneities.	Provide predictive tools that can be employed prospectively in the clinic to design personalized-nanomedicine regimens.
		(Laomettachit et al. 2017)	NP toxicity to healthy liver cells is dose-dependent, and while the effect of small exposures can be reversed due to cell proliferation, tissue damage due to higher dose exposures are generally irreversible.	Such studies are critical in assessing the toxicity potential of nanocarries in clinical doses, thereby providing guidelines for safe exposure limits.